Fermat worked on number theory while preparing an edition of Diophantus, and the notes and comments thereon contained the numerous theorems of considerable elegance necessary to develop the theory of numbers.

Fermat is famous for his "Enigma" that was an extension of Pythagorean Theorem, also known as Fermat's last theorem, which baffled mathematicians for more than 300 years, and was only finally proven in 1994.

Fermat was born at Beaumont-de-Lomagne, 58 kilometers (36 miles) north-west of Toulouse, France.

Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime.

One common explanation is that Fermat made a computational mistake and was so convinced of the correctness of his claim that he failed to double-check his work.

Then n is a Fermat prime if and only if for every a coprime to n, a is a primitive root mod n if and only if a is a quadratic nonresidue mod n.

en.wikipedia.org /wiki/Fermat_number (1063 words)

PlanetMath: Fermat's last theorem(Site not responding. Last check: 2007-11-07)

Fermat's last theorem was actually a conjecture and remained unproved for over 300 years.

We cannot imagine how Fermat's last theorem could be proved without these advanced mathematical tools, which include group theory and Galois theory, the theory of modular forms, Riemannian topology, and the theory of elliptic equations.

Assuming Fermat's teaser was truthful, and Fermat was not in error, this ``paradox'' has lead some to (hopefully) jokingly attribute supernatural abilities to Fermat.

planetmath.org /encyclopedia/FermatsLastTheorem.html (582 words)

Pierre de Fermat(Site not responding. Last check: 2007-11-07)

Pierre de Fermat was born on August 17, 1960, in Beaumont-de-Lomagne, a small town near Toulouse in the south part of France, near the border with Spain.

Fermat's choice of a legal career was natural and typical of his time, for his father's wealth and his mother's famil y background.

"From this principle Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence(in refraction)is a constant number times the sine of the angle of refraction in passing from one medium to anot her." (Bell, p.63).

Fermat's meteoric rise through the government is evidenced by his multiple appointments between 1631 and 1653.

Fermat received a copy of Descartes' "La Dioptrique" from mathematician Beaugrand, but paid it little attention since he was in the middle of a correspondence with Roberval and Etienne Pascal over methods of integration and using them to find centers of gravity.

He wrote to Fermat praising his work on determining the tangent to a cycloid (which is indeed correct), meanwhile writing to Mersenne claiming that it was incorrect and calling Fermat an inadequate mathematician and a thinker.

AW: Fermat was a 17th-century mathematician who wrote a note in the margin of his book stating a particular proposition and claiming to have proved it.

So the romance of Fermat, which had held me all my life, was now combined with a problem that was professionally acceptable.

My wife had heard of Fermat's Last Theorem, but at that time she had no idea of the romantic significance it had for mathematicians, that it had been such a thorn in our flesh for so many years.

www.pbs.org /wgbh/nova/proof/wiles.html (2546 words)

Learn more about Fermat's last theorem in the online encyclopedia.(Site not responding. Last check: 2007-11-07)

The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet's translation of famous Diophantus' Arithmetica, "I have discovered a truly remarkable proof but this margin is too small to contain it".

The methods used by Wiles were unknown when Fermat was writing, and it seems inconceivable that Fermat managed to derive all the necessary mathematics to demonstrate the same solution (in the words of Andrew Wiles, "it's impossible; this is a 20th century proof").

The fact that Fermat never published an attempted proof, or even publicly announced that he had one, suggests that he may have found his own error and simply neglected to cross out his marginal note.

Fermat also obtained the areas of parabolas and hyperbolas of any order, and determined the centres of mass of a few simple laminae and of a paraboloid of revolution.

Fermat's solution depends on the theory of combinations, and will be sufficiently illustrated by the following example, the substance of which is taken from a letter dated August 24, 1654, which occurs in the correspondence with Pascal.

Fermat does not seem to have carried the matter much further, but his correspondence with Pascal shows that his views on the fundamental principles of the subject were accurate: those of Pascal were not altogether correct.

From his appointment on 14 May 1631 Fermat worked in the lower chamber of the parliament but on 16 January 1638 he was appointed to a higher chamber, then in 1652 he was promoted to the highest level at the criminal court.

Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect), that there are exactly two integer solutions of x

Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the form 4k+1 could be written as the sum of two squares.

Fermat is a computer algebra system for Macintosh, Windows, Linux, and Unix by me, Robert H. Lewis of Fordham University, that does arithmetic of arbitrarily long integers and fractions, symbolic calculations, matrices over polynomial rings, graphics, and other numerical calculations.

Fermat has the ability to be interrupted and then later return to the computation, picking up where it left off.

www.bway.net /~lewis (2306 words)

Fermat's Fallibility(Site not responding. Last check: 2007-11-07)

Fermat discussed the propositition that "every number of the form 2^2^n + 1 is a prime" in several letters to several different people (including Frenicle, Pascal, Huygens, Brouckner, Wallis, etc) over a period of many years.

Fermat persisted in his conjecture to the end of his days, usually adding that he had no full proof for it (cf Fe II, 309-310.)" In Dickson's "History of the Theory of Numbers" we find "Fermat expressed his belief that every F_n is a prime, but admitted that he had no proof.

Anyway, it's worth noting that Fermat first worked on this problem in 1640, and as late as June 1658 he was admitting to Brouckner and Wallis that he had no proof (according to Dickson's references).